1. **Problem:** Find the union of sets $A$ and $B$. **Step 1:** Recall the union formula: $$A \cup B = \{x : x \in A \text{ or } x \in B\}$$ **Step 2:** List all unique elements from both sets $A = \{7,9,10,11,12\}$ and $B = \{4,6,10,11,12\}$. **Step 3:** Combine without duplicates: $$A \cup B = \{4,6,7,9,10,11,12\}$$ 2. **Problem:** Find the intersection of sets $A$ and $B$. **Step 1:** Recall the intersection formula: $$A \cap B = \{x : x \in A \text{ and } x \in B\}$$ **Step 2:** Identify common elements in both sets: $10, 11, 12$. **Step 3:** Write the intersection: $$A \cap B = \{10,11,12\}$$ 3. **Problem:** Find the difference $A \setminus B$. **Step 1:** Recall the difference formula: $$A \setminus B = \{x : x \in A \text{ and } x \notin B\}$$ **Step 2:** Elements in $A$ but not in $B$ are $7, 9$. **Step 3:** Write the difference: $$A \setminus B = \{7,9\}$$ 4. **Problem:** Find the difference $B \setminus A$. **Step 1:** Recall the difference formula: $$B \setminus A = \{x : x \in B \text{ and } x \notin A\}$$ **Step 2:** Elements in $B$ but not in $A$ are $4, 6$. **Step 3:** Write the difference: $$B \setminus A = \{4,6\}$$ 5. **Problem:** Find $\overline{B} \cap A$ where $\overline{B}$ is the complement of $B$ in the universal set $U = \{1,2,3,\ldots,12\}$. **Step 1:** Find complement of $B$: $$\overline{B} = U \setminus B = \{1,2,3,5,7,8,9\}$$ **Step 2:** Find intersection with $A$: $$\overline{B} \cap A = \{7,9\}$$ 6. **Problem:** Find $A \setminus \overline{B}$. **Step 1:** Recall $\overline{B} = \{1,2,3,5,7,8,9\}$. **Step 2:** Elements in $A$ but not in $\overline{B}$ are those in $A$ and in $B$: $$A \setminus \overline{B} = A \cap B = \{10,11,12\}$$ 7. **Problem:** Find $A \setminus \overline{A}$. **Step 1:** Complement of $A$ is $$\overline{A} = U \setminus A = \{1,2,3,4,5,6,8\}$$ **Step 2:** Elements in $A$ but not in $\overline{A}$ are all elements of $A$ (since $A$ and $\overline{A}$ are disjoint): $$A \setminus \overline{A} = A = \{7,9,10,11,12\}$$ 8. **Problem:** Find $A \cap \overline{A}$. **Step 1:** Since $A$ and $\overline{A}$ are complements, they have no common elements: $$A \cap \overline{A} = \{\}$$ **Final answers:** 1. $\{4,6,7,9,10,11,12\}$ 2. $\{10,11,12\}$ 3. $\{7,9\}$ 4. $\{4,6\}$ 5. $\{7,9\}$ 6. $\{10,11,12\}$ 7. $\{7,9,10,11,12\}$ 8. $\{\}$